A latent dimension of Condorcet's jury theorem for multiple AI advisers
Kazutoshi Sasahara, Aoi Naito, Ryo Fujie
cs.CY · cs.AI · cs.HC · cs.MA
Quick verdict
Views Condorcet's jury theorem from the user's vantage: dissent and reliability rates cross at p=4/5. Elegant exact theory; validation and code missing.
Summary
The paper revisits Condorcet's jury theorem from the user's rather than the system's vantage. Where the classical theorem tracks how majority reliability Pagg approaches certainty as independent advisers are added, the authors ask how often users will see the panel visibly split. Under a binomial base model (K independent advisers, common accuracy 1/2<p<1, binary verdicts), they derive a closed form for the probability Pvis of a split with at least two dissenters per side and compare exponential convergence rates: reliability converges at (1/2)log(1/(4pq)), visible dissent at log(1/p). The rates cross at adviser accuracy p=4/5. Below that value a finite panel size K*(p) exists where splits become at least as likely as a correct majority (K*=6 at p=0.6, 14 at p=0.7, 32 at p=0.75 under fair tie-breaking); above it, an analytic proof rules out any crossover. Supplementary material extends the analysis to fractional split thresholds, heterogeneous Poisson-binomial panels, and Gaussian-copula correlation, showing correlation suppresses splits and inflates all-wrong unanimity. The split is positioned as a common predictive substrate for predictive multiplicity, reconciliation load, and reliance miscalibration — the latter two linked only through an explicitly untested monotone-response hypothesis. Main limitations: purely analytical and synthetic evidence, no empirical panel data, and code promised but not yet released.
Review complete
Assessment at a glance
4
risk flags identified
● high 0
● medium 2
● low 2
4 strengths · 4 weaknesses · 5 author questions
Levels reflect verification complexity — not a quality score. Evolve never rates your paper.
Key claims under review
- Under the binomial base model, the probability of a visible split (at least two dissenters per side) is exactly Pvis(K,p)=1-q^(K-1)(Kp+q)-p^(K-1)(Kq+p) and approaches 1 for every p<1 as the panel grows.
- The exponential convergence rates of aggregate reliability and visible dissent cross at adviser accuracy p=4/5; below it a finite disclosure reference scale K*(p) exists where splits become at least as likely as a correct majority, and above it an analytic proof rules out any crossover.
- Disagreement among ideal (independent, competent) advisers does not by itself indicate aggregation failure, since most splits occur while the majority is correct (e.g., p=0.9, K=20: majority correct above 99.999%, split seen about 61% of the time).
- The vote split provides a common predictive substrate for predictive multiplicity (exact under the model) and, under an untested monotone-response hypothesis, for reconciliation load and reliance miscalibration.
Strengths
Exact, self-contained theory with an intrinsic-bound argument
Pvis(K,p)=1-q^(K-1)(Kp+q)-p^(K-1)(Kq+p) follows from binomial tails alone (Suppl. Note 2), with domain checks (K>=4), strict monotonicity in K, and side-blind symmetry Pvis(K,p)=Pvis(K,q). Suppl. Note 4 proves no split definition excluding unanimity can place its rate boundary above p=4/5 (since 1-Pvis >= p^K forces Rvis <= log(1/p)), elevating the constant from an observation to a structural property.
Robustness work beyond the headline result
The r-fraction phase diagram (Suppl. Fig. 1), invariance to fixed residual weighting, parity quantification at K* (at p=0.735 ordering holds at K=22, 24 but not 23), and the Poisson-binomial heterogeneity analysis with Supplementary Table 1 (equal mixture of p=0.51/0.89 items reverses the ordering at K=14: E[Pagg]=0.7646 vs E[Pvis]=0.7320, opposite to homogeneous p=0.7) show the boundary claims are not artifacts of one convention.
Clean separation of theorem from interpretation
Numerical and diagnostic hygiene
Weaknesses
Zero empirical validation
Code not available at review time
Idealization gap for the motivating domain
Modest mathematical depth; convention-sensitive K*
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Scope & Claims
This theory paper (arXiv:2609.14438, cs.CY/cs.AI) re-reads Condorcet's jury theorem from the user's vantage. Where the classical theorem tracks aggregate reliability Pagg — the probability the majority verdict is correct as K independent advisers of accuracy p are added — the authors define and track visible dissent Pvis: the probability the user sees at least two votes on each side of a binary question. Central claims: (i) Pvis(K,p) = 1 - q^(K-1)(Kp+q) - p^(K-1)(Kq+p) exactly, tending to 1 for all p<1; (ii) the exponential convergence rates cross at p=4/5 (Ragg = (1/2)log(1/(4pq)) vs Rvis = log(1/p)); (iii) below 4/5 a finite disclosure reference scale K*(p) exists — 6, 14, 32 at p = 0.6, 0.7, 0.75 under fair tie-breaking — above which splits are at least as likely as a correct majority; above 4/5 an analytic proof rules out any crossover; (iv) the split is a common substrate for predictive multiplicity, reconciliation load, and reliance miscalibration.
The remaining sections cover Methodology Assessment, Statistical Evidence, Results & Reproducibility, Limitations & Risks, and Verdict & Recommendations.
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- 4 risk flags with severity, section & reviewer questions
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Frequently asked questions
What does the 4/5 threshold in this paper mean?
If each AI adviser on a panel is individually correct with probability below 0.8 on a binary question, then as more advisers are added the panel eventually becomes more likely to visibly split than to be collectively correct. At or above 0.8 accuracy, a correct majority always stays more likely than a split. The constant comes from comparing the exponential rates at which majority errors and visible splits vanish as panel size grows.
Does disagreement between AI models mean their answer is wrong?
Not necessarily. The paper shows that even panels of independent, competent advisers become near-certain to display a split as the panel grows, and most splits occur while the majority is still correct: at adviser accuracy 0.9 and 20 advisers, the majority is correct more than 99.999% of the time, yet users see a split about 61% of the time.
What is K*, the disclosure reference scale?
K*(p) is the first panel size at which a visible split becomes at least as likely as a correct majority. Under the paper's fair tie-breaking convention, K* equals 6 at adviser accuracy 0.6, 14 at 0.7, and 32 at 0.75; no finite K* exists at accuracy of 0.8 or above.
Is the code or data for this paper available?
There is no empirical data (the work is theory plus simulation). The authors state that accompanying scripts reproduce the numerical results but are not yet publicly available, with a repository link promised in a later version. The mathematical derivations are fully specified in the seven supplementary notes, so the core numbers can be re-derived independently.
About this review
This is an AI-generated preliminary review by Evolve.Pub, produced from the openly available text of the paper. It is not peer review and does not represent the authors' views. The canonical record of the paper is at the publisher link above.
- Figures are available only through extracted captions and in-text descriptions; contour and curve values (Fig. 2d, Fig. 3b, Suppl. Figs. 1-2) could not be inspected directly.
- Several pivotal comparisons cite 2025-2026 preprints (refs. 20, 32, 38, 41-45, 52-54) that could not be verified from the text.
- The extracted text appears complete, including all seven supplementary notes and declarations, so no truncation was detected; Box 1 formatting was reconstructed from linear text.
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